Chi-square distribution

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In probability theory and statistics, the chi-square distribution (also chi-squared or \chi^2  distribution) is one of the most widely used theoretical probability distributions. It is used in statistical significance tests. It is useful, because it is relatively easy to show that certain probability distributions come close to it, under certain conditions. One of these conditions is that the null hypothesis must be true. Another one is that the different random variables (or observations) must be independent of each other.

chi-square
Probability density function
Chi-square distributionPDF.png
Cumulative distribution function
Chi-square distributionCDF.png
Parameters k > 0\, degrees of freedom
Support x \in [0; +\infty)\,
Probability density function (pdf) \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,
Cumulative distribution function (cdf) \frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,
Mean k\,
Median approximately k-2/3\,
Mode k-2\, if k\geq 2\,
Variance 2\,k\,
Skewness \sqrt{8/k}\,
Excess kurtosis 12/k\,
Entropy \frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)
Moment-generating function (mgf) (1-2\,t)^{-k/2} for 2\,t<1\,
Characteristic function (1-2\,i\,t)^{-k/2}\,

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